Laplace Transform for signal and system

1. BVM
ENGINEERING COLLEGE
LAPLACE TRANSFORM
AND TRANSFER
FUNCTIONPresented by:-
Kishan Karangiya (140080111027)
Kishan Mayani (140080111028)
Jay Mistry (140080111029)

2. Definition of Laplace Transform
 The transform method is used to solve certain
problems, that are difficult to solve directly.
 In this method the original problems is first
transformed and solved.
 Laplace transform is one of the tools for solving
ordinary linear differential equations.
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3.  First : Convert the given differential equation
from time domain to complex frequency domain
by taking Laplace transform of the equation
 From this equation, determine the Laplace
transform of the unknown variable
 Finally, convert this expression into time domain
by taking inverse Laplace transform
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4.  Laplace transform method of solving differential
equations offers two distinct advantages over
classical method of problem solving
 From this equation, determine the Laplace
transform of the unknown variable
 Finally, convert this expression into time domain
by taking inverse Laplace transform
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5.  The Laplace transform is defined as below:
Let f(t) be a real function of a real variable t
defined for t>0, then
 Where F(s) is called Laplace transform of f(t).
And the variable ‘s’ which appears in F(s) is
frequency dependent complex variable
 It is given by,
where  = Real part of complex variable ‘s’
 = Imaginary part of complex
variable ‘s’
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  dte.(t)ff(t)LF(s) st-
0



jωσs 

6. Inverse Laplace Transform
 The operation of finding out time domain
function f(t) from Laplace transform F(s) is
called inverse Laplace transform and denoted
as L-1
 Thus,
 The time function f(t) and its Laplace transform
F(s) is called transform pair
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    f(t)(f(t))LLF(s)L -1-1


7. Properties of Laplace Transforms
 The properties of Laplace transform enable us
to find out Laplace transform without having to
compute them directly from the definition.
 The properties are given:
 A) The Linear Property
 The Laplace transformation is a linear operation
– for functions f(t) and g(t), whose Laplace
transforms exists, and constant a and b, the
equation is :
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  bLg(t)aLf(t)g(t)bf(t)aL 

8.  B) Differentiation
 According to this property,
 It means that inverse Laplace transform of a
Laplace transform multiplied by s will give
derivative of the function if initial conditions are
zero.
 C) n-fold differentiation
 According to this property,
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f(0)-sLf(t)
dt
df(t)
L 
(0)f...-(0)fs-f(0)s-Lf(t)s
dt
f(t)d
L 1-n'2-n1-nn
n
n


9.  D) Integration Property
 In general, the Laplace transform of an order n
is
 Laplace transform exists if f(t) does not grow too
fast as
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s
)0(f
Lf(t)
s
1
)(L
1
0








t
f 
 
s
)0(f
...
s
)0(f
s
)0(f
Lf(t)
s
1
f(t)dt....L
n
1-n
2-n
n
1-n
n
n

 
t

10.  E) Time Shift
 The Laplace transform of f(t) delayed by time T
is equal to the Laplace transform of f(t)
multiplied by e-sT ; that is
L[f ( t – T ) u( t – T )] = e-sT F(s), where u (t – T)
denotes the unit step function, which is shifted
to the right in time by T.
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11.  F) Convolution Integral
 The Laplace transform of the product of two
functions F1(s) and F2(s) is given by the
convolution integrals
where L-1F1(s) = f1(t) and L-1F2(s) = f2(t)
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 


d)(f)-(tf
)d-t(ft)(fs)(s)F(FL
2
t
0
1
2
t
0
121
1-





12.  G) Product Transformation
 The Laplace transform of the product of two
functions f1(t) and f2(t) is given by the complex
convolution integral
 H) Frequency Scaling
 The inverse Laplace transform of the functions
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  )d(F)(F
j2
1
t)(t)f(fL 2
jc
j-c
121
1-







f(t)F(s)Lwhereaf(at),
a
s
F 1-







13.  I) Time Scaling
 The Laplace transform of a functions
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Lf(t)F(s)whereaF(as)
a
t
fL 












14.  K) Initial Value Theorem
 The Laplace transform is very useful to find the
initial value of the time function f(t). Thus if F(s)
is the Laplace transform of f(t) then,
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 The only restriction is that f(t) must be
continuous or at the most , a step discontinuity
at t=0.

15.  L) Final Value Theorem
 Similar to the initial value, the Laplace transform is
also useful to find the final value of the time function
f(t).
 Thus if F(s) is the Laplace transform of f(t) then the
final value theorem states that,
 The only restriction is that the roots of the
denominator polynomial of F(s) i.e poles of F(s)
have negative or zero real parts
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16. Table of Laplace Transforms:
Table 1 : Standard Laplace Transform
pairs
f(t) F(s)
1 1/s
Constant K K/s
K f(t), K is constant K F(s)
t 1/s2
tn n/sn+1
e-at 1/s+a
eat 1/s-a
e-at tn n/((s+a)n+1 )
sin t /(s2 + 2)
cos t s/(s2 + 2)
e-at sin t /((s+a)2 + 2)
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17. f(t) F(s)
e-at cos t (s+a)/((s+a)2 + 2)
sinh t /(s2 - 2)
cosh t s/(s2 - 2)
t e-at 1/(s+a)2
1 - e-at a/s(s+a)
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18. Table 2 : Laplace transforms of standard
time functions
Function f(t) Laplace Transform F(s)
Unit step = u(t) 1/s
A u(t) A/s
Delayed unit step = u(t-T) e-Ts/s
A u(t-T) Ae-Ts /s
Unit ramp = r(t) = t u(t) 1/s2
At u(t) A/s2
Delayed unit ramp = r(t-T) = (t-T) u(t-
T)
e-Ts /s2
A(t-T) u(t-T) Ae-Ts /s 2
Unit impulse = (t) 1
Delayed unit impulse = (t-T) e-Ts
Impulse of strength K i.e K (t) K
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19. Inverse Laplace Transform
 Let F(s) is the Laplace transform of f(t) then the
inverse Laplace transform is denoted as,
 The F(s), in partial fraction method, is written in the
form as,
 Where N(s) = Numerator polynomial in s
 D(s) = Denominator polynomial in s
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 F(s)Lf(t) -1

D(s)
N(s)
F(s) 

20. Simple and Real Roots
 The roots of D(s) are simple and real
 The function F(s) can be expressed as,
where a, b, c… are the simple and real roots
of D(s).
 The degree of N(s) should be always less than
D(s)
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c)...-b)(s-a)(s-(s
N(s)
D(s)
N(s)
F(s) 

21.  This can be further expressed as,
where K1, K2, K3 … are called partial fraction
coefficients
 The values of K1, K2, K3 … can be obtained as,
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....
c)-(s
K
b)-(s
K
a)-(s
K
c)...-b)(s-a)(s-(s
N(s)
F(s) 321

cs
bs
as






F(s)c).-(sK
F(s)b).-(sK
F(s)a).-(sK
3
2
1

22.  In general, and so on
 Where sn = nth root of D(s)

 Is standard Laplace transform pair.
 Once F(s) is expressed in terms of partial
fractions, with coefficients K1, K2 … Kn, the
inverse Laplace transform can be easily
obtained
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nss
 F(s).)s-(sK nn
  a)(s
1
eL at


  ...eKeKeKF(s)Lf(t) ct
3
bt
2
at
1
-1


23. Application of Laplace Transform in
Control System
 The control system can be classified as
electrical, mechanical, hydraulic, thermal and so
on.
 All system can be described by
integrodifferential equations of various orders
 While the o/p of such systems for any i/p can be
obtained by solving such integrodifferential
equations
 Mathematically, it is very difficult to solve such
equations in time domain
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24.  The Laplace transform of such integrodifferential
equations converts them into simple algebraic
equations
 All the complicated computations then can be
easily performed in s domain as the equations
to be handled are algebraic in nature.
 Such transformed equations are known as
equations in frequency domain
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25.  By eliminating unwanted variable, the required
variable in s domain can be obtained
 By using technique of Laplace inverse, time
domain function for the required variable can be
obtained
 Hence making the computations easy by
converting the integrodifferential equations into
algebraic is the main essence of the Laplace
transform
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